An electric current is simply the flow of electric charge. In metal wires, electrons make up the flow of charge because they have the ability to move while protons don't. Protons stay in a fixed position because they compose the nuclei of atoms. So what gets electrons moving is a voltage source. Voltage applies pressure or a push on the electrons to move them through a wire. An example of a voltage source is a common battery, because batteries provide a push on electrons.
Electric circuits occur when an electron simply has a path to follow. A continuous flow of electrons occurs when a circuit (a path) has no gaps. An example of an electric circuit includes the wiring in your own home. A wire runs to a bulb and back to the voltage source, which allows the bulb to light your room. Voltage produces current if the circuit is complete.
Getting back to the bird, why didn't he fry? Well, after understanding currents and circuits, I can now answer the question. The bird didn't fry because he didn't complete the circuit. When a circuit is complete and electrons can flow from one end of the voltage source to the other, the circuit is complete and the bird has the circuit running through him. In other words, the bird is like a bulb connected to the circuit in your home, and he too lights up!
In order for the bird to fry he has to complete the circuit so a current can pass through his body. The current is what causes damage to the bird's organs, and tissues. While perched on the wire, every part of the bird's body is at the same voltage as the wire so he feels no effect. Also, when the bird just sits on one wire, he feels nothing because he has not completed a circuit. If the bird puts one foot on each wire, he feels the current because he has completed the circuit, allowing for electrons to flow through him. Voltage produces a current only when a circuit is complete.
Source: Conceptual Physics 8th Edition. Paul G. Hewitt. Addison-Wesley. Book.
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